One of 12 books developed for use with the core material (Book O) of the Elements of Mathematics Program, this text covers material well beyond the scope of the usual secondary mathematics sequences. These materials are designed for highly motivated students with strong verbal abilities; mathematical theories and ideas are developed through problem situations and explanatory materials. This volume examines the formal system of plane geometry, in particular, the parallel postulate. Affine and projective geometries are then developed. Discussions of the history of Euclidean geometry, the parallel postulate, and set theory are included. (SD)